Proof: We follow here closely the proof of [1]}, adapting it to the equivariant case. The map \(\Phi :U\rightarrow V\)  is a \(\mathbb {C}^\ast \) -equivariant embedding of smooth schemes of respective dimension \(\dim (U)=m,\dim (V)=m+n\) . We can then find \(\mathbb {C}^\ast \) -equivariant étale coordinates \((\dot{y}_1,...,\dot{y}_m,\dot{z}_1,...,\dot{z}_n)\)  on a \(\mathbb {C}^\ast \) -equivariant Zariski open neighborhood \(\dot{V}\)  of \(j(x)\)  in \(V\)  such that \(j(x) = (0,..., 0)\)  and \(\Phi (U)\cap \dot{V}\)  is the locus \(\dot{z}_1=...=\dot{z}_n=0\)  in \(\dot{V}\) . Set \(\dot{U}=\Phi ^{-1}(\dot{V})\)  and \(\dot{x}_a=\dot{y}_a\circ \Phi |_{\dot{U}}\)  for \(a = 1, . . . , m\) . Then \(\dot{U}\)  is an open neighborhood of \(i(x)\)  in \(U\)  , and \((\dot{x}_1, ...,\dot{x}_m)\)  are étale coordinates on \(\dot{U}\)  with \(i(x) = (0,...,0)\) . Then the ideal \(I_{R,U}=I_{(df)}\)  is on \(\dot{U}\)  the ideal generated by the \(\mathbb {C}^\ast \) -equivariant functions \(\frac{\partial f}{\partial \dot{x}_a}\)  for \(a = 1, . . . , m\) , and the ideal \(I_{S,V}= I_{(dg)}\)  is on \(\dot{V}\)  the ideal generated by the \(\mathbb {C}^\ast \) -equivariant functions \(\frac{\partial g}{\partial \dot{y}_a}\)  for \(a = 1, . . . , m\)  and \(\frac{\partial g}{\partial \dot{z}_b}\)  for \(b = 1, . . . , n\) . Since \(\Phi \)  maps \(U\)  to \(\dot{z}_1=...=\dot{z}_n=0\)  and \(i(R)\)  to \(j(R)\subset j(S)\) , we have \(I_{(df)}\simeq I_{(dg)} |_{\dot{z}_1=...=\dot{z}_n=0}\)  , that is:
\((\frac{\partial f}{\partial \dot{x}_a}(\dot{y}_1,...,\dot{y}_m):a=1,...,m)=(&\frac{\partial g}{\partial \dot{y}_a}(\dot{y}_1,...,\dot{y}_m,0,...,0):a=1,...,m,\nonumber \\&\frac{\partial g}{\partial \dot{z}_b}(\dot{y}_1,...,\dot{y}_m,0,...,0):b=1,...,n)\) 
